To solve the inequality \( q^2 \leq 3 \), we first rewrite it in a form that can be analyzed.
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Determine the roots: Start by setting the inequality to zero:
\[ q^2 - 3 \leq 0 \]
This can be factored or solved by finding the square roots:
\[ q^2 = 3 \implies q = \pm \sqrt{3} \]
Thus, the critical points are \( q = -\sqrt{3} \) and \( q = \sqrt{3} \).
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Test intervals: The critical points divide the real number line into intervals. We need to test the inequality in the intervals \( (-\infty, -\sqrt{3}) \), \( (-\sqrt{3}, \sqrt{3}) \), and \( (\sqrt{3}, \infty) \).
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Interval \( (-\infty, -\sqrt{3}) \): Choose \( q = -2 \) \[ (-2)^2 = 4 \quad \text{(not }\leq 3\text{)} \]
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Interval \( (-\sqrt{3}, \sqrt{3}) \): Choose \( q = 0 \) \[ 0^2 = 0 \quad \text{(which is } \leq 3\text{)} \]
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Interval \( (\sqrt{3}, \infty) \): Choose \( q = 2 \) \[ 2^2 = 4 \quad \text{(not }\leq 3\text{)} \]
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Conclusion: The inequality is satisfied in the interval \( (-\sqrt{3}, \sqrt{3}) \) and at the endpoints.
The solution to the inequality is:
\[ [-\sqrt{3}, \sqrt{3}] \]
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Graphing the solution:
- Plot the points \( -\sqrt{3} \) and \( \sqrt{3} \):
- Since both endpoints are included, we will denote them with closed circles.
- The middle part between the two endpoints should be shaded.
- Plot the points \( -\sqrt{3} \) and \( \sqrt{3} \):
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Adjusting the plot: If we select an endpoint to change from closed to open, we could choose, for example, \( -\sqrt{3} \) to be open. The final graphical representation will look as follows:
- Plot:
- Open circle at \( -\sqrt{3} \) (not included)
- Closed circle at \( \sqrt{3} \) (included)
- Shade the interval between these two points.
This represents the solution to the inequality \( q^2 \leq 3 \) with the specified adjustments.
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